Method of controlling growth of a semiconductor crystal to automatically transition from taper growth to target diameter growth

ABSTRACT

A control method for use with a Czochralski crystal puller. The method includes pulling the growing crystal from the melt at a first target pull rate to grow a taper portion of the crystal and measuring the crystal diameter of the taper. The method also includes estimating a slope of the diameter as a function of a change in crystal diameter relative to time and the first target pull rate. The method further includes predicting a crystal diameter D i  at which to initiate body growth from the taper as a function of the estimated slope. By increasing the pull rate to a second target pull rate when the measured crystal diameter reaches the predicted crystal diameter D i , the method controls growth of the crystal for transitioning from taper growth to body growth. The method also determines the second target pull rate as a function of the estimated slope when using a predefined diameter D i  at which to initiate growth of the crystal body.

BACKGROUND OF THE INVENTION

This invention relates generally to improvements in controlling growthprocesses of single crystal semiconductors for use in the manufacture ofelectronic components and, particularly, to a method for accuratelycontrolling growth in a Czochralski crystal growth process fortransitioning from taper growth to target diameter growth.

Monocrystalline, or single crystal, silicon is the starting material inmost processes for fabricating semiconductor electronic components.Crystal pulling machines employing the Czochralski process produce themajority of single crystal silicon. Briefly described, the Czochralskiprocess involves melting a charge of high-purity polycrystalline siliconin a quartz crucible located in a specifically designed furnace. Afterthe heated crucible melts the silicon charge, a crystal liftingmechanism lowers a seed crystal into contact with the molten silicon.The mechanism then withdraws the seed to pull a growing crystal from thesilicon melt.

After formation of a crystal neck, the growth process enlarges thediameter of the growing crystal in a cone-shaped manner by decreasingthe pulling rate and/or the melt temperature until a desired diameter isreached. This portion of the crystal is typically referred to as thecrown or taper. By controlling the pull rate and the melt temperaturewhile compensating for the decreasing melt level, the main body of thecrystal is grown so that it has an approximately constant diameter(i.e., it is generally cylindrical). Near the end of the growth processbut before the crucible is emptied of molten silicon, the processgradually reduces the crystal diameter to form an end cone. Typically,the end cone is formed by increasing the crystal pull rate and heatsupplied to the crucible. When the diameter becomes small enough, thecrystal is then separated from the melt. During the growth process, thecrucible rotates the melt in one direction and the crystal liftingmechanism rotates its pulling cable, or shaft, along with the seed andthe crystal, in an opposite direction.

Although presently available Czochralski growth processes have beensatisfactory for growing single crystal silicon useful in a wide varietyof applications, further improvements are still desired. For example, itis desired to provide more accurate transitions from taper growth to thebody target diameter.

The conventional method for transitioning from taper growth to bodygrowth involves increasing the crystal lift rate. This causes a changein the rate of diameter increase, from some positive value to nearlyzero, or even a slightly negative value. The conventional transitioningmethod intends to arrive at a steady-state diameter value foressentially straight crystal growth that is equal to a crystal targetdiameter. Currently, this transition occurs at a fixed, predeterminedtaper diameter regardless of the conditions inside the crystal grower.In the alternative, an operator decides when to initiate the transition.Unfortunately, the differing experience levels of various operators, inaddition to differing thermal conditions inside the crystal grower,produce different taper growth rates. For this reason, the conventionalmethods for initiating the transition to body growth often producediffering results from one crystal growth run to another. In oneinstance the initial crystal body may be grown with too small of adiameter but the initial body may be grown with too large of a diameterin another. Specifically, there is often a relatively large standarddeviation in crystal diameter in the early body growth compared to thedesired target diameter for the crystal. This requires correction by thecontrol system during the remainder of the body growth. Moreover, if acrystal is unacceptably undersized in its early body growth, thensignificant portions will be unusable for semiconductor waferfabrication.

For these reasons, an accurate and reliable apparatus and method forcontrolling silicon crystal growth to automatically transition fromtaper growth to target diameter growth is desired.

SUMMARY OF THE INVENTION

The invention meets the above needs and overcomes the deficiencies ofthe prior art by providing a method for automatically transitioning fromtaper growth to target diameter growth in a crystal ingot pulled from amelt according to the Czochralski process. Among the several objects ofthe invention may be noted the provision of such method that permitsmore accurate taper to body transitions; the provision of such methodthat provides repeatable results; the provision of such method thatsignificantly lowers the initial diameter standard deviation betweencrystals; the provision of such method that predicts the diameter atwhich to begin the transition to straight crystal growth; the provisionof such method that may be incorporated into the controls of an existingcrystal pulling device; and the provision of such method that can becarried out efficiently and economically.

Briefly described, a control method embodying aspects of the inventionis for use with a crystal puller for growing a monocrystallinesemiconductor crystal according to the Czochralski process. The crystalpuller has a heated crucible containing a semiconductor melt from whichthe crystal is grown. The crystal is grown on a seed crystal pulled fromthe melt. The method includes the step of pulling the growing crystalfrom the melt at a first target pull rate. The first target pull ratesubstantially follows an initial velocity profile for growing a taperportion of the crystal. In the taper portion, the crystal has agenerally increasing diameter. The method also includes measuring thecrystal diameter of the taper and estimating a slope of the diameter.The estimated slope is a function of a change in crystal diameterrelative to time and the first target pull rate. The method furtherincludes the step of predicting a crystal diameter measurement D_(i) atwhich to initiate shouldering as a function of the estimated slope.After shouldering, the body of the crystal has a substantially uniformdiameter greater than the predicted shouldering initiation diametermeasurement D_(i). By increasing the pull rate by an increment k to asecond target pull rate when the measured crystal diameter reaches thepredicted crystal diameter measurement D_(i), the method utilizes thenatural response of the crystal plus the measurement bias for moreaccurate transitioning from taper growth to body growth.

Another embodiment of the invention is directed to a control method foruse with a crystal puller for growing a monocrystalline semiconductorcrystal according to the Czochralski process. The crystal puller has aheated crucible containing a semiconductor melt from which the crystalis grown. The crystal is grown on a seed crystal pulled from the melt.The method includes the step of pulling the growing crystal from themelt at a first target pull rate. The first target pull ratesubstantially follows an initial velocity profile for growing a taperportion of the crystal. In the taper portion, the crystal has agenerally increasing diameter. The method also includes measuring thecrystal diameter of the taper and estimating a slope of the diameter.The estimated slope is a function of a change in crystal diameterrelative to time and the first target pull rate. The method furtherincludes the step of predefining a crystal diameter measurement D_(i) atwhich to initiate a transition to a body portion of the crystal from thetaper. The body of the crystal has a substantially uniform diametergreater than the predefined diameter measurement D_(i). The method alsoincludes determining an increment of the pull rate that corresponds toan accurate transition into body growth as a function of the estimatedslope and one or more hotzone parameters. By increasing the pull rate tothe second target pull rate when the measured crystal diameter reachesthe predefined crystal diameter measurement D_(i), the method utilizesthe natural response of the crystal plus the measurement bias for moreaccurate transitioning from taper growth to body growth.

Alternatively, the invention may comprise various other methods andsystems.

Other objects and features will be in part apparent and in part pointedout hereinafter.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a partially schematic, block diagram illustrating a crystalpuller and an apparatus according to the present invention forcontrolling the crystal puller.

FIG. 2 is a block diagram of a control unit of the system of FIG. 1.

FIG. 3 is a schematic, fragmentary view of taper growth of asemiconductor crystal being pulled from a melt contained in the crystalpuller of FIG. 1.

FIG. 4 is graph of exemplary data representing a diameter slope modelfitting.

FIG. 5 is a schematic, fragmentary view of a bright ring of thesemiconductor crystal being pulled from the melt contained in thecrystal puller of FIG. 1.

FIG. 6 is graph of exemplary data representing diameter bias versustangent of diameter slope.

Corresponding reference characters indicate corresponding partsthroughout the drawings.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

Referring now to FIG. 1, a control apparatus, indicated generally at 11,is shown for use with a Czochralski crystal growing apparatus, indicatedgenerally at 13. The details of construction of the crystal growingapparatus, or puller, 13 are well known to those of ordinary skill inthe art. In general, the crystal puller 13 includes a vacuum chamber 15enclosing a crucible 19. Heating means such as a resistance heater 21surrounds the crucible 19. In one embodiment, insulation 23 lines theinner wall of vacuum chamber 15 and a chamber cooling jacket (not shown)fed with water surrounds it. A vacuum pump (not shown) typically removesgas from within the vacuum chamber 15 as an inert atmosphere of argongas is fed into it.

According to the Czochralski single crystal growth process, a quantityof polycrystalline silicon, or polysilicon, is charged to crucible 19. Aheater power supply 27 provides electric current through the resistanceheater 21 to melt the charge and, thus, form a silicon melt 29 fromwhich a single crystal 31 is pulled. As is known in the art, the singlecrystal 31 starts with a seed crystal 35 attached to a pull shaft, orcable, 37. As shown in FIG. 1, single crystal 31 and crucible 19generally have a common axis of symmetry 39. During both heating andcrystal pulling, a crucible drive unit 43 rotates crucible 19 (e.g. inthe clockwise direction). The crucible drive unit 43 also raises andlowers crucible 19 as desired during the growth process. For example,crucible drive unit 43 raises crucible 19 as the melt 29 is depleted tomaintain its level, indicated by reference character 45, at a desiredheight. A crystal drive unit 47 similarly rotates the cable 37 and,thus, rotates crystal 31, in a direction opposite the direction in whichcrucible drive unit 43 rotates crucible 19. In addition, the crystaldrive unit 47 raises and lowers crystal 31 relative to the melt level 45as desired during the growth process. The details of construction ofcrucible drive unit 43 and crystal drive unit 47 are well known to thoseof ordinary skill in the art.

In one embodiment, crystal puller 13 first preheats the seed crystal 35by lowering it nearly into contact with the molten silicon of melt 29contained by crucible 19. After preheating, crystal drive unit 47continues to lower seed crystal 35 via cable 37 into contact with melt29 at its melt level 45. As seed crystal 35 melts, crystal drive unit 47slowly withdraws, or pulls, it from the melt 29. Seed crystal 35 drawssilicon from melt 29 to produce a growth of silicon single crystal 31 asit is withdrawn. Crystal drive unit 47 rotates crystal 31 at a referencerate as it pulls crystal 31 from melt 29. Crucible drive unit 43similarly rotates crucible 19 at another reference rate, but usually inthe opposite direction relative to crystal 31.

A control unit 51 initially controls the withdrawal, or pull, rate andthe power that power supply 27 provides to heater 21 to cause a neckdown of crystal 31. Preferably, crystal puller 13 grows the crystal neckat a substantially constant diameter as seed crystal 35 is drawn frommelt 29. For example, the control unit 51 maintains a substantiallyconstant neck diameter of about five percent of the desired bodydiameter. Under a conventional control scheme, control unit 51 adjuststhe rotation, pull and/or heating parameters after the neck reaches adesired length to cause the diameter of crystal 31 to increase in acone-shaped manner until a desired crystal body diameter is reached. Forexample, control unit 51 decreases the pull rate according to an initialvelocity profile to create an outwardly flaring region typicallyreferred to as the taper of the crystal. Commonly assigned applicationSer. No. 09/287,916, filed Apr. 7, 1999, the entire disclosure of whichis incorporated herein by reference, discloses a closed loop method andsystem for accurately controlling taper growth in a Czochralski crystalgrowth process.

As described in detail below, control unit 51 calculates the slope ofthe taper (and hence, knowing the mean seed lift rate, the diametergrowth rate may be computed) using an estimate of the rate of change ofthe diameter slope. Based on these calculations, control unit 51predicts a desired crystal diameter for initiating the transition fromtaper growth to body growth. Control unit 51 then controls the growthparameters to maintain a relatively constant diameter as measured byapparatus 11 until the process approaches its end. At that point, thepull rate and heating are usually increased for decreasing the diameterto form an end cone, i.e., a tapered portion at the end of singlecrystal 31. Once the diameter of the end cone is sufficiently small(e.g., 2 mm to 4 mm), detachment of crystal 31 from melt 29 can beaccomplished without causing dislocations to spread to the main body ofcrystal 31. Crystal 31 is then removed from vacuum chamber 15 forprocessing into wafers. Commonly assigned U.S. Pat. No. 5,178,720, theentire disclosure of which is incorporated herein by reference,discloses a preferred method for controlling crystal and cruciblerotation rates as a function of the crystal diameter.

Preferably, control unit 51 operates in combination with at least onetwo-dimensional camera 53 to determine a plurality of parameters of thegrowth process including the crystal diameter. Camera 53 is mountedabove a viewport (not shown) of chamber 15 and aimed generally at theintersection of longitudinal axis 39 and melt level 45 (see FIG. 5). Inaddition to processing signals from camera 53, control unit 51 processessignals from other sensors. For example, a temperature sensor 59, suchas a radiation pyrometer, may be used to measure the melt surfacetemperature.

FIG. 2 illustrates a preferred embodiment of control unit 51 in blockdiagram form. Control unit 51 includes a programmed digital or analogcomputer 61 for use in controlling, among other things, crucible driveunit 43, single crystal drive unit 47 and heater power supply 27 as afunction of processed signals from camera 53 and other sensors. As shownin FIG. 2, a programmable logic controller (PLC) 63 communicates withthe computer 61 via line 67 (e.g., RS-232 cable) and with one or moreprocess input/output modules 69 via line 71 (e.g., RS-485 cable).According to the invention, computer 61 provides an operator interfacethat permits the operator of the crystal growing apparatus 13 to input aset of desired parameters for the particular crystal being grown.

The process input/output module 69 provides a path to and from crystalpuller 13 for controlling the growth process. As an example, the PLC 63receives information regarding the melt temperature from temperaturesensor 59 and outputs a control signal to heater power supply 27 viaprocess input/output module 69 for controlling the melt temperaturethereby controlling the growth process.

Referring further to FIG. 2, camera 53 communicates video images of theinterior of crucible 19 via line 77 (e.g., RS-170 video cable) to avision system 79 providing edge detection and diameter measurementcalculations. In turn, vision system 79 communicates with PLC 63 vialine 83. In one preferred embodiment, PLC 63 is a Model TI 575 PLCmanufactured by Siemens or a Model 545 PLC manufactured by TexasInstruments and line 83 represents a communications interface (e.g., VMEbackplane interface). Depending on the particular controller embodyingPLC 63, the communications interface 83 may be, for example, a customVME rack that includes an additional communications board (e.g., Model2571 Program Port Expander Module using the RS-422 serial bidirectionalPLC port).

FIG. 3 illustrates a relatively early phase of the crystal growthprocess following melt-down and dipping of seed crystal 35. Afterformation of a crystal neck 85, the typical process enlarges thediameter of the growing crystal 31 by decreasing the pulling rate and/orthe melt temperature until a desired diameter is reached. This portionof increasing diameter is referred to as a taper, or crown, 87. As thetaper 87 increases to a desired diameter, crystal 31 forms a shoulder,indicated generally at 91, followed by a main body (not shown in FIG.3). As the crystal drive unit 47 pulls crystal 31 from melt 29, a liquidmeniscus 95 forms at the interface between crystal 31 and melt 29. Theliquid meniscus 95 forms on a surface 97 of melt 29. As is known in theart, the reflection of crucible 19 on meniscus 95 is often visible as abright ring adjacent crystal 31.

It should be understood that the as-grown crystal body typically willnot have an entirely uniform diameter, although it is generallycylindrical. For this reason, the diameter of crystal 31 may varyslightly at different axial positions along axis 39. Further, thediameter varies in the different phases of crystal growth (i.e., seed,neck, taper, body and end cone). Although described in connection withvision system 79, it is to be further understood that crystal diametermeasurements may be determined by a number of methods. For example,several technologies are known for providing crystal diametermeasurements including methods that claim to measure the width of thebright ring. The bright ring is a characteristic of the reflection ofthe crucible wall in the meniscus that is formed at the solid-liquidinterface of crystal 31. Conventional bright ring and meniscus sensorsemploy optical pyrometers, photocells, rotating mirrors with photocells,light sources with photocells, line-scan cameras, and two-dimensionalarray cameras. Commonly assigned U.S. Pat. Nos. 5,665,159 and 5,653,799,the entire disclosures of which are incorporated herein by reference,describe a system and method, respectively, for accurately and reliablymeasuring crystal diameter for use in controlling the growth process ofsingle crystal silicon. Advantageously, the system and method of thesepatents accurately determine the growing crystal's diameter byprocessing images of the crystal-melt interface generated by a camera.

According to a preferred embodiment of the present invention, PLC 63processes seed lift rate and taper diameter data to calculate the slopeof taper 87. This information yields data representing the crystalgrowth rate. In addition, PLC 63 calculates the rate of change of thetaper slope. Based on these calculations, PLC 63 advantageously predictsa crystal diameter at which transition from taper growth to body growthis desired based on the current values of the diameter and diameterslope computation and, thus, based on the current conditions in puller13. This gives repeatable, more accurate transitions from taper 87 tothe body of crystal 31, and significantly lowers the initial diameterstandard deviation.

In this embodiment, PLC 63 executes routines to compute an optimalshoulder roll, requiring minimal control changes. To derive the shoulderinitiation equation, first assume that the diameter slope responds witha well known first order system plus delay model. By first ignoring thedelay, a state-space representation of the Laplace Transform processmodel may be generated:$\frac{D(s)}{V_{P}(s)} = \left. \frac{G_{P}}{s\left( {{s\quad \tau} + 1} \right)}\Rightarrow{\begin{Bmatrix}{{\overset{.}{x}(t)} = {{{\begin{bmatrix}0 & 1 \\0 & {{- 1}/\tau}\end{bmatrix}{x(t)}} + {\begin{bmatrix}0 \\{1/\tau}\end{bmatrix}{V_{P}(t)}}} \equiv {{{Ax}(t)} + {{BV}_{P}(t)}}}} \\{{D(t)} = {{\left\lbrack {G_{P}\quad 0} \right\rbrack {x(t)}} = {{Cx}(t)}}}\end{Bmatrix}.} \right.$

By the variation of constants (VOC) formula taught by C. T. Chen, LINEARSYSTEM THEORY AND DESIGN 1-227 (Oxford University Press) (1984),x(t) = Φ(t, t₀)x(t₀) + ∫_(t₀)^(t)Φ(t, ξ)BV_(p)  ξ,

where:${\Phi \left( {t,\xi} \right)} = {^{A{({t - \xi})}} = \begin{bmatrix}1 & {\tau \left( {1 - ^{\frac{- {({t - \xi})}}{\tau}}} \right)} \\0 & ^{\frac{- {({t - \xi})}}{\tau}}\end{bmatrix}}$

by series expansion, inverse Laplace transform (e^(At)=L⁻¹{(sI−A)⁻¹}),or other methods. For an open-loop predictive shoulder roll, it isdesirable to increment the pull rate V_(P) by a constant value, namely,k, such that the desired target diameter D_(t) is reached with zeroslope. Even if the loop is closed, a computed set point for V_(P) duringshouldering may be desired. During crown growth, a shoulder initiationdiameter D_(i) is measured periodically, with a slope {dot over (D)}_(i)interpreted. If the desired final slope is zero, then the VOC formulaprovides the relation: $\begin{matrix}{{\frac{1}{G_{P}}\left\{ {\begin{bmatrix}D_{t} \\0\end{bmatrix} - {\begin{bmatrix}1 & {\tau \left( {1 - ^{\frac{- T}{\tau}}} \right)} \\0 & ^{\frac{- T}{\tau}}\end{bmatrix}\begin{bmatrix}D_{i} \\{\overset{.}{D}}_{i}\end{bmatrix}}} \right\}} = {\int_{0}^{T}{\begin{bmatrix}{1 - {\tau \left( {1 - ^{\frac{- {({T - \xi})}}{\tau}}} \right)}} \\{\frac{1}{\tau}^{\frac{- {({T - \xi})}}{\tau}}}\end{bmatrix}\quad k{{\xi}.}}}} \\{Since} \\{{{\int_{0}^{T}{^{\frac{- {({T - \xi})}}{\tau}}\quad {\xi}}} = {{{{\tau }^{\frac{- {({T - \xi})}}{\tau}}}}_{0}^{T} = {\tau \left( {1 - ^{\frac{- T}{\tau}}} \right)}}},}\end{matrix}$

the relation noted above yields two formulas, namely:${{- \frac{{\overset{.}{D}}_{i}}{G_{P}}}^{\frac{- T}{\tau}}} = {k\left( {1 - ^{\frac{- T}{\tau}}} \right)}$and${G_{P}{k\left\lbrack {T - {\tau \left( {1 - ^{\frac{- T}{\tau}}} \right)}} \right\rbrack}} = {\left\lbrack {D_{t} - D_{i} - {{\tau \left( {1 - ^{\frac{- T}{\tau}}} \right)}{\overset{.}{D}}_{i}}} \right\rbrack.}$

Assuming that T=−τln(Z), simplifying the first formula gives:$T = {{- \tau}\quad {\ln \left( \frac{k}{k - {{\overset{.}{D}}_{i}/G_{P}}} \right)}}$

after solving for, and then eliminating, Z. Substituting this value of Tinto the second equation above results in an estimated diameter forinitiating the transition from taper growth to body growth. In otherwords, control unit 51 preferably increases the pull rate set point tothe predetermined value k when the diameter of crystal 31 reaches D_(i),defined by:$D_{i} = {D_{t} - {\tau {\left\{ {{\overset{.}{D}}_{i} + {\left\lbrack {{G_{P}k} - {\overset{.}{D}}_{i}} \right\rbrack \left( \frac{k}{k - {{\overset{.}{D}}_{i}/G_{P}}} \right)} - {G_{P}{k\left\lbrack {1 + {\ln \left( \frac{k}{k - {{\overset{.}{D}}_{i}/G_{P}}} \right)}} \right\rbrack}}} \right\}.}}}$

In this instance, k represents an amount of increase in the pull rate toachieve the desired shoulder roll, τ is the process time constant inminutes and G_(P) is the DC process gain in (mm diameter/min)/(mm/minpull rate change).

According to a preferred embodiment of the invention, PLC 63 calculatesthe estimated diameter D_(i) for initiating the shoulder roll inresponse to a predetermined pull rate value of k. It is to be understoodthat PLC 63 could likewise calculate a desired value of k for selectingthe seed lift set point based on a predetermined shoulder transitioninitiation diameter. In either case, the present invention providesoptimal shoulder roll with minimal changes to the control schemeexecuted by control unit 51.

Although the use of the above equation provides improved results intransitioning from taper growth to body growth, the present inventionalso accounts for two additional factors in the shoulder roll, namely,the growth of the bright ring during shouldering and any pure time delayassociated with the diameter response. PLC 63 preferably uses an apriori approximation to account for the bright ring around crystal 31. Adiameter bias DB associated with the formation of the bright ring isapproximated by DB=max(0,α₀+α₁tanθ+α₂tan²θ), where 0 is the angle of thecrown growth relative to vertical (see FIG. 3). First, noting that DB=α₀for vertical body growth,ΔDB=DB_(taper)−DB_(body)=max(−α₀,α₁tanθ+α₂tan²θ), where ΔDB is theamount that the crown appears smaller than the body diameter whencompared with the body diameter. Substituting the relation:${\tan \quad \theta} = {{\frac{1}{2V_{P}}\frac{D}{t}} = \frac{\overset{.}{D}}{2V_{P}}}$

yields the following approximation for use in compensating for changesin the width of the bright ring as a function of the diameter slope:${\Delta \quad {DB}} = {{\max \left( {{- \alpha_{0}},{{\frac{\alpha_{1}}{2}\left( \frac{\overset{.}{D}}{V_{P}} \right)} + {\frac{\alpha_{2}}{4}\left( \frac{\overset{.}{D}}{V_{P}} \right)^{2}}}} \right)}.}$

Appendix A provides a description of bright ring modeling according to apreferred embodiment of the invention for use in determining thediameter bias of the bright ring. In Appendix A, a polynomial equationis used to approximate the bright ring. Appendix B provides an exemplaryprogram for use with Matlab® software for performing the calculations ofAppendix A. Preferably, control unit 51 executes the program to performnumerical searching to calculate tanθ to enable an approximation of thebright ring with a quadratic. It is to be understood that other softwaremay be used for performing this function (e.g., Mathcad® or Lotus®software).

The PLC 63 further improves the estimated diameter for initiating thetransition by accounting for pure time delay with a time delayapproximation. This provides a more accurate model of the process. Ingeneral, the time delay represents the amount of time required followinga control change before any change in the output can be detected.Specifically, the time delay represents the time that the crown maycontinue to grow at the same rate after a change in the pull rate setpoint is made until the change begins to take effect. In a model suchas:${\frac{D(s)}{V_{P}(s)} = \frac{G_{P}^{- {st}_{d}}}{s\left( {{s\quad \tau} + 1} \right)}},$

the parameter t_(d) is a pure time delay, such that no change in thebright ring diameter slope measurement occurs until t_(d) after a changein the pull rate. In this case, the diameter of crystal 31 grows byΔD≈{dot over (D)}_(i)t_(d) during the time delay, and the shoulderinitiation point is therefore to be reduced by this quantity for a moreaccurate final diameter.

Summing each of the above components gives a final formula for theoptimum shoulder roll initiation, given a desired increase in pull rateduring shouldering represented by k, a current measure of the diameterD_(i), and a current estimate of the diameter slope {dot over (D)}_(i):$D_{i} = {D_{t} - {\tau \left\{ {{\overset{.}{D}}_{i} + {\left\lbrack {{G_{P}k} - {\overset{.}{D}}_{i}} \right\rbrack \left( \frac{k}{k - {{\overset{.}{D}}_{i}/G_{P}}} \right)} - {G_{P}{k\left\lbrack {1 + {\ln \left( \frac{k}{k - {{\overset{.}{D}}_{i}/G_{P}}} \right)}} \right\rbrack}}} \right\}} + {\max \left( {{- \alpha_{0}},{{\frac{\alpha_{1}}{2}\left( \frac{\overset{.}{D}}{V_{P}} \right)} + {\frac{\alpha_{2}}{4}\left( \frac{\overset{.}{D}}{V_{P}} \right)^{2}}}} \right)} - {{\overset{.}{D}}_{i}{t_{d}.}}}$

Note that when crown/shoulder data is used to generate the values ofG_(P), τ, and t_(d) that the bright ring function should first besubtracted from the measured diameter data to provide more accurateprocess response measurements. Additionally, the bright ring, being areflection of the bright wall of crucible 19 on the reflective meniscus,varies by hotzone configuration, melt level and the like, and should bemodeled uniquely for such variations.

Referring now to FIG. 4, a data set taken from approximately 20 priorruns of a 300 mm crystal growing process provided data regarding growthin the shoulder region for estimating the process dynamics. The dataindicates a diameter slope response to pull during shoulder growth(i.e., the transition from taper to body). In this instance, the data isused to define the response characteristics, that is, to determineminimum mean squared error parameter values of τ=9.5 min., G_(P)=−5.3,and t_(d)=4 min.

It is contemplated that the estimated diameter slope {dot over({circumflex over (D)})}_(i) may be generated using a well known bestlinear estimate. In one embodiment, the present invention uses aquadratic diameter growth model for the taper diameter to generate atime-based estimate using a time series of diameter measurements. Thediameter approximation is a polynomial so the slope can be calculated bytaking the derivative of the approximation.

Following implementation with PLC 63, the predictive shoulder method ofthe present invention incorporates the following variables into adecision formula that decides whether the diameter is large enough toinitiate shouldering: 1) a statistical estimation of the diameter slopeduring the last stages of crown growth; 2) filtered pull rate values (ifactive crown slope control is used); 3) a shouldering seed lift (aconstant recipe value); 4) a meniscus diameter adjustment as a functionof diameter slope (for a given hotzone design); 5) a current diametermeasurement; and 6) estimated process dynamics (time delay, timeconstant, and gain).

Statistics taken from the historical data indicates initial bodydiameters from −6 mm below set point to +4.5 mm above set point, with astandard deviation of 5.63 mm. The use of the predictive shoulder rollsoftware of the present invention resulted in a sample standarddeviation of 1.2 on eight shoulders. This represents an approximately4.5 times decrease in variability. Advantageously, such improved resultsare achieved without any “tuning factors” being required. That is, theshoulder trigger point is based on real-time process data and priorcomputations alone.

In view of the above, it will be seen that the several objects of theinvention are achieved and other advantageous results attained.

As various changes could be made in the above constructions and methodswithout departing from the scope of the invention, it is intended thatall matter contained in the above description or shown in theaccompanying drawings shall be interpreted as illustrative and not in alimiting sense.

APPENDIX A

Referring now to FIG. 5, as crystal drive unit 47 pulls crystal 31 frommelt 29, the liquid meniscus 95 forms at the interface between crystal31 and melt 29. Liquid meniscus 95 forms on the surface 97 of melt 29.As is known in the art, the reflection of crucible 19 on meniscus 95 isoften visible as a bright ring adjacent crystal 31. In this diagram:$\theta_{C} \approx {\tan^{- 1}\left( \frac{2\left( {h_{C} - {MD}} \right)}{D_{C} - D} \right)}$and${{\pi/2} + \theta_{V} - \theta_{R}} = {\theta_{C} + {\theta_{R}.{{Simplifying}:{\theta_{R} \approx {{\frac{1}{2}\left\lbrack {\frac{\pi}{2} + \theta_{V} - {\tan^{- 1}\left( \frac{2\left( {h_{C} - {MD}} \right)}{D_{C} - D} \right)}} \right\rbrack}.}}}}}$

D. T. J. Hurle, Analytical Representation of the Shape of the Meniscusin Czochralski Growth, 63 Journal of Crystal Growth 13-17 (1983),describes an approximate analytical relationship for representing theshape of the meniscus and its dependence on the angle of contact at thethree-phase boundary and on the crystal radius. By using the separabledifferential equation provided by the reference:${{{\left\lbrack {1 + \left( \frac{y}{x} \right)^{2}} \right\rbrack \left\lbrack {1 - {ay}^{2}} \right\rbrack}^{2} = 1};{a = {\frac{1}{2}{\left( \frac{2}{\beta} \right)^{3/2}\left\lbrack {1 + {\frac{\beta}{2}\frac{\sin \quad \alpha^{0}}{rh}}} \right\rbrack}}}},$

and the relation y={square root over ((1+L −cos θ_(R)+L )/a)}, themeniscus bias E_(D) may be computed, or:

(D,MD,h _(C),θ_(V),θ_(S))θ_(R)(y,h)[E_(D)=2(x−r)].

Using published values of these parameters with dimensions from thehotzone, one can fit a quadratic function to the diameter error versusthe diameter slope. For example, using the values shown in Table I,below:

TABLE I D D_(c) hc-MD β 310 mm 380 mm 30 mm 58 mm²

gives the circular data points in FIG. 6, using Hurle's solution:$x = {\frac{D}{2} + \sqrt{\frac{2}{a} - h^{2}} - \sqrt{\frac{2}{a} - y^{2}} - {\frac{1}{\sqrt{2a}}{\ln \left\lbrack {\left( \frac{y}{h} \right)\left( \frac{\sqrt{2 - {ah}^{2}}}{\sqrt{2 - {ay}^{2}}} \right)} \right\rbrack}}}$

where h={square root over (β(1+L −cos α°)+(β sin α°/4+L r)²+L )}−β sinα°/4r represents the meniscus height and α°=π/2−θ_(L)°−θ,θ_(L)°=11° forsilicon.

In FIG. 6, the angle θ represents the diameter slope of taper 87 (e.g.,θ=0 for straight vertical growth). As shown, the data points can befitted to a polynomial function, in this case a quadratic, yielding abright ring width DB in terms of the crown slope θ. For this specificmelt level and set of process parameters the quadratic estimate is givenby: DB=−2.997(tan θ)²−2.543 tan θ+5.84

APPENDIX B

Matlab® Program:

% M-file edest.m, which computes, given hotzone dimensions, % the errorin diameter estimate.

beta—58;

thetaL0=11;

hc=30;

T=0;

Dc=380;

MD=0;

D=310;

thetaV=atan((400−D/2)/750);

thetaR=(pi/2+thetaV−atan(2*(hc−MD−T)/(Dc−D)))/2;

thetaSmin=0;thetaSmax=60;stepthetaS=1;

Rc=ceil((thetaSmax-thetaSmin+1)/stepthetaS);

edest=zeros(Rc,3);

for thetaS=thetaSmin:1:thetaSmax

alpha0=pi/180*(90−thetaL0−thetaS);

h=sqrt(beta*(1−cos(alpha0))+(beta*sin(alpha0)/2/D){circumflex over ()}2)−beta* sin(alpha0)/2/D;

a=(1/beta+sin(alpha0)/D/h);

y=fmin(‘menslope’,0.001,h,[ ],a,thetaR);

x=D/2+sqrt(2/a−h{circumflex over ( )}2)−sqrt(2/a−y{circumflex over ()}2)−log(y/h*(sqrt(2)+sqrt(2−a*h{circumflex over ()}2))/(sqrt(2)+sqrt(2−a*y{circumflex over ( )}2)))/sqrt(2*a);

if 2*x−D<0.001

x=D/2;

end

edest(thetaS−thetaSmin+1,:)=[thetaS,tan(pi/180*thetaS),2*x−D];

end

clf;subplot(111);plot(edest(:,2),edest(:,3),‘ro’)

fitter=edest(min(find(edest(:,3))):max(find(edest(:,3))),:);

[P,S]=polyfit(fitter(:,2),fitter(:,3),2);

format short e;P,format;

predvals=polyval(P,fitter(:,2));maxerror=max(predvals-fitter(:,3))

hold on;plot(fitter(:,2),predvals,‘b−’);hold off;

grid on;title(‘300mm process, Dc=310 mm, Hr−MD=50 mm’);ylabel(‘2×BrightRing Width’);xlabel(‘tan(theta)’);

gtext(‘−2.997*(tan(theta)){circumflex over ()}2−2.543*tan(theta)+5.84’).

What is claimed is:
 1. A control method for use with a crystal pullerfor growing a monocrystalline semiconductor crystal according to theCzochralski process, said crystal puller having a heated cruciblecontaining a semiconductor melt from which the crystal is grown, saidcrystal being grown on a seed crystal pulled from the melt, said methodcomprising the steps of: pulling the growing crystal from the melt at afirst target pull rate, said first target pull rate substantiallyfollowing an initial velocity profile for growing a taper portion of thecrystal, said taper portion of the crystal having a generally increasingdiameter; measuring the crystal diameter during the taper portion ofgrowth; estimating a slope of the crystal diameter, said estimated slopebeing a function of a change in crystal diameter relative to time duringthe taper portion of growth and the first target pull rate; predicting acrystal diameter measurement D_(i) at which to initiate shouldering forgrowing a body portion of the crystal from the taper portion of thecrystal, said shouldering initiation diameter measurement D_(i) being afunction of the estimated slope, said body portion of the crystal havinga substantially uniform diameter greater than the predicted diametermeasurement D_(i); and increasing the pull rate to a second target pullrate when the measured crystal diameter reaches the shoulderinginitiation diameter measurement D_(i) for controlling a transition fromthe taper portion of the crystal to the body portion of the crystal,said second target pull rate corresponding to initial growth of the bodyportion of the crystal.
 2. The method of claim 1 wherein the secondtarget pull rate is a substantially constant, predetermined rate.
 3. Themethod of claim 1 wherein the step of predicting the shoulderinginitiation diameter measurement D_(i) includes calculating Di accordingto the following equation:$D_{i} = {D_{t} - {\tau \left\{ {{\overset{.}{D}}_{i} + {\left\lbrack {{G_{P}k} - {\overset{.}{D}}_{i}} \right\rbrack \left( \frac{k}{k - {{\overset{.}{D}}_{i}/G_{P}}} \right)} - {G_{P}{k\left\lbrack {1 + {\ln \left( \frac{k}{k - {{\overset{.}{D}}_{i}/G_{P}}} \right)}} \right\rbrack}}} \right\}}}$

where D_(t) is a final target diameter of the body portion of thecrystal, {dot over (D)}_(i) is the estimated slope of the crystaldiameter, k is a planned increment in the pull rate for shouldering, τis a process time constant and G_(P) is a DC process gain.
 4. The methodof claim 1 wherein the melt has a surface with a meniscus visible as abright area adjacent the crystal as the crystal is pulled from the meltand wherein the step of predicting the shouldering initiation diametermeasurement D_(i) includes compensating for a diameter bias resultingfrom the width of the bright ring.
 5. The method of claim 1 wherein thestep of predicting the shouldering initiation diameter measurement D_(i)includes calculating D_(i) according to the following equation:$D_{i} = {D_{t} - {\tau \left\{ {{\overset{.}{D}}_{i} + {\left\lbrack {{G_{P}k} - {\overset{.}{D}}_{i}} \right\rbrack \left( \frac{k}{k - {{\overset{.}{D}}_{i}/G_{P}}} \right)} - {G_{P}{k\left\lbrack {1 + {\ln \left( \frac{k}{k - {{\overset{.}{D}}_{i}/G_{P}}} \right)}} \right\rbrack}}} \right\}} + {\Delta \quad {DB}} - {{\overset{.}{D}}_{i}t_{d}}}$

where D_(t) is a final target diameter of the body portion of thecrystal, {dot over (D)}_(i) is the estimated slope of the crystaldiameter, k is a planned increment in the pull rate for shouldering, τis a process time constant, G_(P) is a DC process gain, ΔDB is adiameter bias approximation and {dot over (D)}_(i)t_(d) is anapproximation of additional diameter growth that occurs during the timedelay.
 6. The method of claim 5 wherein the melt has a surface with ameniscus visible as a bright area adjacent to the crystal as the crystalis pulled from the melt and further comprising the step of defining thediameter bias approximation based on a model of the width of the brightring.
 7. The method of claim 6 further comprising the step of definingthe diameter bias approximation according to the equation:${\Delta \quad {DB}} = {{\max \left( {{- \alpha_{0}},{{\frac{\alpha_{1}}{2}\left( \frac{\overset{.}{D}}{V_{P}} \right)} + {\frac{\alpha_{2}}{4}\left( \frac{\overset{.}{D}}{V_{P}} \right)^{2}}}} \right)}.}$

where {dot over (D)}_(i) is the estimated slope of the crystal diameter,V_(P) is the pull rate and α₀, α₁ and α₂ are constants determined from amathematical model approximating the bright ring width.
 8. The method ofclaim 6 further comprising the step of modeling the bright ring as apolynomial equation, said equation being a function of the slope of thetaper portion of the crystal, and the step of defining the diameter biasapproximation as a function of a width of the bright ring as modeled bythe equation.
 9. The method of claim 1 wherein the step of predictingthe shouldering initiation diameter measurement D_(i) includescalculating D_(i) as a function of one or more of the followingvariables: a statistical estimation of the diameter slope during growthof the taper portion of the crystal; a desired pull rate increment; asubstantially constant shouldering pull rate; a meniscus bright ringwidth adjustment as a function of diameter slope; a current diametermeasurement; and estimated process dynamics.
 10. The method of claim 1wherein the step of estimating the diameter slope includes modeling thediameter slope as a first order system plus delay.
 11. The method ofclaim 1 wherein the step of estimating the diameter slope includesgenerating a polynomial model of the growth of the crystal based on dataincluding a time series of diameter measurements and performing a bestlinear estimate of the data.
 12. The method of claim 1 furthercomprising the steps of recording diameter measurements from a pluralityof prior runs of the crystal puller and hotzone configuration anddetermining a set of response characteristics corresponding to thecrystal puller and wherein the step of estimating the diameter slopeincludes defining the estimated diameter slope as a function of theresponse characteristics.
 13. A control method for use with a crystalpuller for growing a monocrystalline semiconductor crystal according tothe Czochralski process, said crystal puller having a heated cruciblecontaining a semiconductor melt from which the crystal is grown, saidcrystal being grown on a seed crystal pulled from the melt, said methodcomprising the steps of: pulling the growing crystal from the melt at afirst target pull rate, said first target pull rate substantiallyfollowing an initial velocity profile for growing a taper portion of thecrystal, said taper portion of the crystal having a generally increasingdiameter; measuring the crystal diameter during the taper portion ofgrowth; estimating a slope of the crystal diameter, said estimated slopebeing a function of a change in crystal diameter relative to time duringthe taper portion of growth and the first target pull rate; predefininga crystal diameter measurement D_(i) at which to initiate a transitionto a body portion of the crystal from the taper portion of the crystal,said body portion of the crystal having a substantially uniform diametergreater than the predefined diameter measurement D_(i); determining asecond target pull rate as a function of the estimated slope and one ormore hotzone parameters, said second target pull rate corresponding togrowth of the crystal to achieve the transition to the body portion ofthe crystal at a target diameter with zero slope; and increasing thepull rate to the second target pull rate when the measured crystaldiameter reaches the predefined crystal diameter measurement D_(i) forcontrolling a transition from the taper portion of the crystal to thebody portion of the crystal.